Our continuing research is based on four general interests and areas of expertise, which run through all our specific aims: (i) reference Bayesian methods, based on prior distributions chosen by some formal role, (ii) model selection and model averaging, (iii) sensitivity to modeling assumptions, and development of more flexible models, and (iv) computational techniques. Our research is aimed at grappling with a series of problems that are at once pressing in practice and fundamental. A continuing objective of our work is to advance the use of Bayesian methods in the biomedical and biobehavioral sciences, particularly in clinical trials, analysis of longitudinal studies, and diagnostic classification. The general methodological results we expect to obtain are motivated in part by our vigorous participation in several cross-disciplinary domains including cancer, mental health, and neuroscience, which we describe briefly in the Methods section. Through application of our methods will also learn about their practical value and discover any limitations they may have. Elaboration of simple parametric models has been a major theme in Statistics in the latter part of this century. One of the successful ideas has been to introduce covariates with regression-like structure into models for non-normal data; another is to specify parameters of interest, but let the remainder of the model remain non-parametric; and a third is to suppose a parameter itself follows a distribution. Our proposed research incorporates the first two ideas in several places, but is concerned primarily with the last notion, involving hierarchical models, mixture models, and latent variable models. With increased computing power, especially using Markov chain Monte Carlo (MCMC) methods, these random-parameter models have become central to much current statistical activity. Yet, despite great progress in the use of these models, fundamental issues remain, as we explain below, this is the main motivation of our work.